TERM PAPER

MECHANICS: PHY101

Topic: CYCLOTRON & BETATRON DOA: 27/08/2010

DOR: 25/09/2010

DOS: 13/11/2010 Submitted to: Submitted by:

Mr. AMANDEEP SINGH BISWAJIT SEN

Deptt. Of Physics Roll.No.A17

Reg.No.11007652

Class RK4002

ACKNOWLEDGEMENT

I take this opportunity to present my votes of thanks to all those guidepost who really acted as lightening pillars to enlighten our way throughout this project that has led to successful and satisfactory completion of this study.

I am highly thankful to Mr. AMANDEEP SINGH for his active support, valuable time and advice, whole-hearted guidance, sincere cooperation and pains-taking involvement during the study and in completing the assignment of preparing the said project within the time stipulated.

Lastly, We are thankful to all those, particularly the various friends , who have been instrumental in creating proper, healthy and conductive environment and including new and fresh innovative ideas for us during the project, their help, it would have been extremely difficult for us to prepare the project in a time bound framework.

Name –BISWAJIT SEN

Regd.No-11007652

Rollno. RK4002A17

CERTIFICATE

Name of school: LITCA

This is to certify that BISWAJIT SEN of section K4002, B.Tech(ME) has satisfactorily

completed the term paper in mechanics of first semester.

Date : 7/11/2010

Registration No. :11007652

Signature of Internal Examiner

Signature of External Examiner

# # Contents

# Cyclotron How the cyclotron works Uses of the cyclotron

Problems solved by the cyclotron

Advantages of the cyclotron

Limitations of the cyclotron

Mathematics of the cyclotron

o Non-relativistic

o Relativistic

Related technologies

# Betatron # How it works

# Applications

# Limitations

# External links

# # # INTRODUCTION

# #Cyclotron

In technology, a cyclotron is a type of particle accelerator. In physics, cyclotron frequency is the frequency of a charged particle moving perpendicularly to the direction of a uniform magnetic field, i.e. a magnetic field of constant magnitude and direction. Since that motion is always circular, the cyclotron frequency is well defined.

Cyclotrons accelerate charged particles using a high-frequency, alternating voltage (potential difference). A perpendicular magnetic field causes the particles to spiral almost in a circle so that they re-encounter the accelerating voltage many times.

The first cyclotron was manufactured by Ernest Lawrence, of the University of California, Berkeley who started operating it in 1932, though others had been working along similar lines at the time. The first European cyclotron was founded in Leningrad in the physics department of the Radium Institute (Head Vitali Khlopin). In 1932 George Gamow and Lev Mysovskii presented a draft for consideration by the Scientific Council of the Radium Institute, and the approval of it, under the guidance and direct participation of the Igor Kurchatov and Lev Mysovskii cyclotron was installed and running by 1937.

The American physicist Ernest O. Lawrence won the 1939 Nobel Prize in physics for a breakthrough in accelerator design in the early 1930s. He developed the cyclotron, the first circular accelerator. A cyclotron is somewhat like a linac wrapped into a tight spiral. Instead of many tubes, the machine has only two hollow vacuum chambers, called dees, that are shaped like capital letter Ds back to back .A magnetic field, produced by a powerful electromagnet, keeps the particles moving in a circle. Each time the charged particles pass through the gap between the dees, they are accelerated. As the particles gain energy, they spiral out toward the edge of the accelerator until they gain enough energy to exit the accelerator.

TRIUMF, Canada's national laboratory for nuclear and particle physics, houses the world's largest cyclotron. The 18m diameter, 4000 tonne main magnet produces a field of 0.46 T while a 23 MHz 94 kV electric field is used to accelerate the 300 μA beam. TRIUMF is run

by a consortium of fifteen Canadian universities and is located at the University of British Columbia, Vancouver, Canada.

# # How the cyclotron works

Diagram of cyclotron operation from Lawrence's 1934 patent.

Beam of electrons moving in a circle. Lighting is caused by excitation of gas atoms in a bulb.

The electrodes shown at the right would be in the vacuum chamber, which is flat, in a narrow gap between the two poles of a large magnet.

In the cyclotron, a high-frequency alternating voltage applied across the "D" electrodes (also called "dees") alternately attracts and repels charged particles. The particles, injected near the center of the magnetic field, accelerate only when passing through the gap between the electrodes. The perpendicular magnetic field (passing vertically through the "D" electrodes) , combined with the increasing energy of the particles forces the particles to travel in a spiral path.

With no change in energy the charged particles in a magnetic field will follow a circular path. In the cyclotron, energy is applied to the particles as they cross the gap between the dees and so they are accelerated (at the typical sub-relativistic speeds used) and will increase in mass as they approach the speed of light. Either of these effects (increased velocity or increased mass) will increase the radius of the circle and so the path will be a spiral.

(The particles move in a spiral, because a current of electrons or ions, flowing perpendicular to a magnetic field, experiences a force perpendicular to its direction of motion. The charged particles move freely in a vacuum, so the particles follow a spiral path.)

The radius will increase until the particles hit a target at the perimeter of the vacuum chamber. Various materials may be used for the target, and the collisions will create secondary particles which may be guided outside of the cyclotron and into instruments for analysis. The results will enable the calculation of various properties, such as the mean spacing between atoms and the creation of various collision products. Subsequent chemical and particle analysis of the target material may give insight into nuclear transmutation of the elements used in the target.

# # Uses of the cyclotron

## For several decades, cyclotrons were the best source of high-energy beams for nuclear physics experiments; several cyclotrons are still in use for this type of research.

## Cyclotrons can be used to treat cancer. Ion beams from cyclotrons can be used, as in proton therapy, to penetrate the body and kill tumors by radiation damage, while minimizing damage to healthy tissue along their path.

## Cyclotron beams can be used to bombard other atoms to produce short-lived positron-emitting isotopes suitable for PET imaging.

# # Problems solved by the cyclotron

60-inch cyclotron, circa 1939, showing a beam of accelerated ions (likely protons or deuterons) escaping the accelerator and ionizing the surrounding air causing a blue glow.

The cyclotron was an improvement over the linear accelerators that were available when it was invented. A linear accelerator (also called a linac) accelerates particles in a straight line through an evacuated tube (or series of such tubes placed end to end). A set of electrodes shaped like flat donuts are arranged inside the length of the tube(s). These are driven by high-power radio waves that continuously switch between positive and negative voltage, causing particles traveling along the center of the tube to accelerate. In the 1920s, it was not possible to get high frequency radio waves at high power, so either the accelerating electrodes had to be far apart to accommodate the low frequency or more stages were required to compensate for the low power at each stage. Either way, higher-energy particles required longer accelerators than scientists could afford.

Modern linacs use high power Klystrons and other devices able to impart much more power at higher frequencies. But before these devices existed, cyclotrons were cheaper than linacs.

Cyclotrons accelerate particles in a spiral path. Therefore, a compact accelerator can contain much more distance than a linear accelerator, with more opportunities to accelerate the particles.

# # Advantages of the cyclotron

Cyclotrons have a single electrical driver, which saves both money and power, since more expense may be allocated to increasing efficiency.

Cyclotrons produce a continuous stream of particles at the target, so the average power is relatively high.

The compactness of the device reduces other costs, such as its foundations, radiation shielding, and the enclosing building.

# # Limitations of the cyclotron

The magnet portion of a 27" cyclotron. The gray object is the upper pole piece, routing the magnetic field in two loops through a similar part below. The white canisters held conductive coils to generate the magnetic field. The D electrodes are contained in a vacuum chamber that was inserted in the central field gap.

The spiral path of the cyclotron beam can only "sync up" with klystron-type (constant frequency) voltage sources if the accelerated particles are approximately obeying Newton's Laws of Motion. If the particles become fast enough that relativistic effects become important, the beam gets out of phase with the oscillating electric field, and cannot receive any additional acceleration. The cyclotron is therefore only capable of accelerating particles up to a few percent of the speed of light. To accommodate increased mass the magnetic field may be modified by appropriately shaping the pole pieces as in the isochronous cyclotrons, operating in a pulsed mode and changing the frequency applied to the dees as in the synchrocyclotrons, either of which is limited by the diminishing cost effectiveness of making larger machines. Cost limitations have been overcome by employing the more complex synchrotron or linear accelerator, both of which have the advantage of scalability, offering more power within an improved cost structure as the machines are made larger.

# # Mathematics of the cyclotron

# # Non-relativistic

The centripetal force is provided by the transverse magnetic field B, and the force on a particle travelling in a magnetic field (which causes it to be angularly displaced, i.e. spiral) is equal to Bqv. So,

(Where m is the mass of the particle, q is its charge, B the magnetic field strength, v is its velocity and r is the radius of its path.)

The speed at which the particles enter the cyclotron due to a potential difference, V.

Therefore,

v/r is equal to angular velocity, ω, so

And since the angular frequency is

ω = 2πf

Therefore,

The frequency of the driving voltage is simply the inverse of this frequency so that the particle crosses between the dees at the same point in the voltage cycle.

A pair of "dee" electrodes with loops of coolant pipes on their surface at the Lawrence Hall of Science. The particle exit point may be seen at the top of the upper dee, where the target would be positioned

This shows that for a particle of constant mass, the frequency does not depend upon the radius of the particle's orbit. As the beam spirals out, its frequency does not decrease, and it must continue to accelerate, as it is travelling more distance in the same time. As particles approach the speed of light, they acquire additional mass, requiring modifications to the frequency, or the magnetic field during the acceleration.

# # Relativistic

The radius of curvature for a particle moving relativistically in a static magnetic field is

where

the Lorentz factor

Note that in high-energy experiments energy, E, and momentum, p, are used rather than velocity, and both measured in units of energy. In that case one should use the substitution,

where this is in Natural units

The relativistic cyclotron frequency is

,wherefc is the classical frequency, given above, of a charged particle with velocity

v circling in a magnetic field.

The rest mass of an electron is 511 keV/c2, so the frequency correction is 1% for a magnetic vacuum tube with a 5.11 keV/c2 direct current accelerating voltage. The proton mass is nearly two thousand times the electron mass, so the 1% correction energy is about 9 MeV, which is sufficient to induce nuclear reactions.

An alternative to the synchrocyclotron is the isochronous cyclotron, which has a magnetic field that increases with radius, rather than with time. The de-focusing effect of this radial field gradient is compensated by ridges on the magnet faces which vary the field azimuthally as well. This allows particles to be accelerated continuously, on every period of the radio frequency (RF), rather than in bursts as in most other accelerator types. This principle that alternating field gradients have a net focusing effect is called strong focusing. It was obscurely known theoretically long before it was put into practice.

# # Related technologies

The spiraling of electrons in a cylindrical vacuum chamber within a transverse magnetic field is also employed in the magnetron, a device for producing high frequency radio waves (microwaves).

# # Betatron

A betatron is a cyclotron developed by Donald Kerst at the University of Illinois in 1940 to accelerate electrons, but the concepts ultimately originate from Rolf Widerøe and previous development occurred in Germany through Max Steinbeck in the 1930s.[1] The betatron is essentially a transformer with a torus-shaped vacuum tube as its secondary coil. An alternating current in the primary coils accelerates electrons in the vacuum around a circular path.

Invented in 1940 by D. W. Kerst, the betatron is a circular accelerator for electrons. It differs from the cyclotron in that the acceleration of the electrons is achieved by increasing the magnetic flux through the orbit of the electrons. The electrons travel in an orbit of fixed radius, and are accelerated by the magnetic field which also keeps them in the orbit.

In the betatron, one must deal with the relativistic effects on the electron as well as the loss of energy from radiation. All accelerated charges radiate electromagnetic energy, and accelerated electrons radiate more energy in a given kinetic energy range than would protons. This radiative energy loss, called synchrotron radiation, limits the maximum energy from the betatron to a few hundred MeV. The acceleration program for the increasing of the magnetic flux to keep the electron accelerating in a fixed-radius orbit must account for the change in the relativistic particle energy mc2.

Unlike other particle accelerators, the name of the betatron does not tell us anything about how it works, but it is best thought of as a transformer, with a ring of electrons as the secondary coil. The magnetic field used to make the electrons move in a circle is also the one used to accelerate them, although the magnet must be carefully designed so that the field strength at the orbit radius is equal to half the average field strength linking the orbit:

If the magnetic field increases, there is a changing flux linking the loop of electrons and so an induced e.m.f. which accelerates the electrons. As the electrons get faster they need a larger magnetic field to keep moving at a constant radius, which is provided by the increasing field – the effects are proportional, so the field is always strong enough to keep the electrons in orbit!

The field is changed by passing an alternating current through the primary coils and particle acceleration occurs on the first quarter of the voltage sine wave’s cycle. Although the last quarter of the cycle also has a changing field that would accelerate the electrons, it is in the wrong direction for them to move in the correct circle! The target is bombarded with pulses of particles at the frequency of the ac supply.

The particles have maximum energy when the magnetic field is at its strongest value but the formula used for the cyclotron will not work for betatrons because the electron will be relativistic. However, if the total energy is much greater than the rest energy then E = pc is a good approximation. As the centripetal force is again provided by the Lorentz force,

the maximum momentum will be and hence

An example of this type of machine is the University of Chicago’s 315 MeV betatron, built in 1949. With a magnetic field strength of 0.92 T and orbit radius of 1.22 m, the calculated energy is 340 MeV, in reasonable agreement, as synchrotron radiation losses have not been taken into account. (see section on synchrotrons)

The formula for the electron’s momentum can also be derived from Faraday’s law of electromagnetic induction.

where N = 1 and A = r2.

Converting from voltage induced to electric field strength using E = V/d gives

and so

 

The force on the electron will be given by so Newton’s second law of motion gives

and integrating gives

 

# # How it works

In a betatron, the magnetic field spins the injected electrons and accelerates them at the center where there is a ring-shaped vacuum tube changing the magnetic field and producing an electric field in the vacuum ring.

The stable orbit for the electrons satisfies where θ0 is the flux with the orbit

at r0 is the radius and H0 is the magnetic field at r0. In other words, the magnetic field at the orbit must be half the average magnetic field over its circular cross section.

# # Applications

Betatrons were historically employed in particle physics experiments to provide high energy beams of electrons—up to about 300 MeV. If the electron beam is directed at a metal plate, the betatron can be used as a source of energetic x-rays or gamma rays; these x-rays may be used in industrial and medical applications (historically in radiation oncology). A small version of a Betatron was also used to provide electrons converted into neutrons by a target to provide prompt initiation of some nuclear weapons.

The Radiation Center, the first private medical center to treat cancer patients with a betatron was opened by Dr. O. Arthur Stiennon, in a suburb of Madison, Wisconsin in the late 1950s.

# # Limitations

The maximum energy that betatron can impart is limited by the strength of the magnetic field due to the saturation of iron and by practical size of the magnet core. The next generation of accelerators, the synchrotrons, overcome these limitations.

# # External links

Cyclotron (google ebook) Topic cyclotron (Britannica , encyclopedia) Topic betatron (answer.com) Topic betatron (Nptel.iitm.ac.in)